Sharp weighted estimates for strong-sparse operators

TL;DR

This paper establishes the optimal weighted $L^2$ bounds for strong-sparse operators by constructing a specific lacunary mixture of dual power weights, demonstrating the sharpness of the known bounds.

Contribution

It introduces a novel weight construction that proves the sharpness of the weighted bounds for strong-sparse operators.

Findings

Sharp weighted $L^2$ bounds are proven for strong-sparse operators.

A lacunary mixture of dual power weights is constructed to demonstrate bound sharpness.

The trivial upper bound of the operator norm is shown to be sharp.

Abstract

We prove the sharp weighted-$L^2$ bounds for the strong-sparse operators introduced in \cite{KaragulyanM}. The main contribution of the paper is the construction of a weight that is a lacunary mixture of dual power weights. This weights helps to prove the sharpness of the trivial upper bound of the operator norm.